Properties

Label 245.2.b.d
Level $245$
Weight $2$
Character orbit 245.b
Analytic conductor $1.956$
Analytic rank $0$
Dimension $2$
CM discriminant -35
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [245,2,Mod(99,245)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(245, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("245.99");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 245 = 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 245.b (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.95633484952\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-5}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{3} + 2 q^{4} + \beta q^{5} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{3} + 2 q^{4} + \beta q^{5} - 2 q^{9} - 3 q^{11} + 2 \beta q^{12} - 3 \beta q^{13} - 5 q^{15} + 4 q^{16} + \beta q^{17} + 2 \beta q^{20} - 5 q^{25} + \beta q^{27} + 9 q^{29} - 3 \beta q^{33} - 4 q^{36} + 15 q^{39} - 6 q^{44} - 2 \beta q^{45} - 5 \beta q^{47} + 4 \beta q^{48} - 5 q^{51} - 6 \beta q^{52} - 3 \beta q^{55} - 10 q^{60} + 8 q^{64} + 15 q^{65} + 2 \beta q^{68} - 12 q^{71} - 6 \beta q^{73} - 5 \beta q^{75} + q^{79} + 4 \beta q^{80} - 11 q^{81} + 4 \beta q^{83} - 5 q^{85} + 9 \beta q^{87} - 3 \beta q^{97} + 6 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{4} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{4} - 4 q^{9} - 6 q^{11} - 10 q^{15} + 8 q^{16} - 10 q^{25} + 18 q^{29} - 8 q^{36} + 30 q^{39} - 12 q^{44} - 10 q^{51} - 20 q^{60} + 16 q^{64} + 30 q^{65} - 24 q^{71} + 2 q^{79} - 22 q^{81} - 10 q^{85} + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/245\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(197\)
\(\chi(n)\) \(1\) \(-1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
99.1
2.23607i
2.23607i
0 2.23607i 2.00000 2.23607i 0 0 0 −2.00000 0
99.2 0 2.23607i 2.00000 2.23607i 0 0 0 −2.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
35.c odd 2 1 CM by \(\Q(\sqrt{-35}) \)
5.b even 2 1 inner
7.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 245.2.b.d 2
3.b odd 2 1 2205.2.d.h 2
5.b even 2 1 inner 245.2.b.d 2
5.c odd 4 2 1225.2.a.q 2
7.b odd 2 1 inner 245.2.b.d 2
7.c even 3 2 245.2.j.b 4
7.d odd 6 2 245.2.j.b 4
15.d odd 2 1 2205.2.d.h 2
21.c even 2 1 2205.2.d.h 2
35.c odd 2 1 CM 245.2.b.d 2
35.f even 4 2 1225.2.a.q 2
35.i odd 6 2 245.2.j.b 4
35.j even 6 2 245.2.j.b 4
105.g even 2 1 2205.2.d.h 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
245.2.b.d 2 1.a even 1 1 trivial
245.2.b.d 2 5.b even 2 1 inner
245.2.b.d 2 7.b odd 2 1 inner
245.2.b.d 2 35.c odd 2 1 CM
245.2.j.b 4 7.c even 3 2
245.2.j.b 4 7.d odd 6 2
245.2.j.b 4 35.i odd 6 2
245.2.j.b 4 35.j even 6 2
1225.2.a.q 2 5.c odd 4 2
1225.2.a.q 2 35.f even 4 2
2205.2.d.h 2 3.b odd 2 1
2205.2.d.h 2 15.d odd 2 1
2205.2.d.h 2 21.c even 2 1
2205.2.d.h 2 105.g even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(245, [\chi])\):

\( T_{2} \) Copy content Toggle raw display
\( T_{3}^{2} + 5 \) Copy content Toggle raw display
\( T_{19} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 5 \) Copy content Toggle raw display
$5$ \( T^{2} + 5 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( (T + 3)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 45 \) Copy content Toggle raw display
$17$ \( T^{2} + 5 \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( (T - 9)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 125 \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} \) Copy content Toggle raw display
$67$ \( T^{2} \) Copy content Toggle raw display
$71$ \( (T + 12)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 180 \) Copy content Toggle raw display
$79$ \( (T - 1)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 80 \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 45 \) Copy content Toggle raw display
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